852 research outputs found
Possible thermodynamic structure underlying the laws of Zipf and Benford
We show that the laws of Zipf and Benford, obeyed by scores of numerical data
generated by many and diverse kinds of natural phenomena and human activity are
related to the focal expression of a generalized thermodynamic structure. This
structure is obtained from a deformed type of statistical mechanics that arises
when configurational phase space is incompletely visited in a severe way.
Specifically, the restriction is that the accessible fraction of this space has
fractal properties. The focal expression is an (incomplete) Legendre transform
between two entropy (or Massieu) potentials that when particularized to first
digits leads to a previously existing generalization of Benford's law. The
inverse functional of this expression leads to Zipf's law; but it naturally
includes the bends or tails observed in real data for small and large rank.
Remarkably, we find that the entire problem is analogous to the transition to
chaos via intermittency exhibited by low-dimensional nonlinear maps. Our
results also explain the generic form of the degree distribution of scale-free
networks.Comment: To be published in European Physical Journal
Predictability: a way to characterize Complexity
Different aspects of the predictability problem in dynamical systems are
reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy,
Shannon entropy and algorithmic complexity is discussed. In particular, we
emphasize how a characterization of the unpredictability of a system gives a
measure of its complexity. Adopting this point of view, we review some
developments in the characterization of the predictability of systems showing
different kind of complexity: from low-dimensional systems to high-dimensional
ones with spatio-temporal chaos and to fully developed turbulence. A special
attention is devoted to finite-time and finite-resolution effects on
predictability, which can be accounted with suitable generalization of the
standard indicators. The problems involved in systems with intrinsic randomness
is discussed, with emphasis on the important problems of distinguishing chaos
from noise and of modeling the system. The characterization of irregular
behavior in systems with discrete phase space is also considered.Comment: 142 Latex pgs. 41 included eps figures, submitted to Physics Reports.
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